Modules over Valuation Rings

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CRC Press, Mar 27, 1985 - Mathematics - 336 pages
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This book initiates a systematic, in-depth study of Modules Over Valuation Domains. It introduces the theory of modules over commutative domains without finiteness conditions and examines frontiers of current research in modules over valuation domains. It represents a unique effort to combine ideas from abelian group theory, in a large scale, with powerful techniques developed in module theory. This volume surveys the background material on valuation rings, modules and homological algebra ... features new results for important classes of modules such as finitely generated, divisible, pure-injective, and projective dimension one -- never published before ... contains exercises and research problems -- offering guidance for independent and creative study ... and provides historical notes, comments, and an extensive bibliography. Mathematicians and advanced graduate-level mathematics students interested in module theory, abelian group theory, and commutative ring theory can stay abreast of the latest advances with Modules Over Valuation Domains. Book jacket.
 

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Contents

VALUATION RINGS
1
PRELIMINARIES ON MODULES
31
HOMOLOGICAL PRELIMINARIES
57
PROJECTIVITY AND PROJECTIVE DIMENSION
70
TOPOLOGY AND FILTRATIONS
94
DIVISIBILITY AND INJECTIVITY
116
UNISERIAL MODULES
140
HEIGHTS AND INDICATORS
156
Cotorsion modules
242
The cotorsion hull
245
Notes Problem 24
248
TORSION MODULES
250
Embedding in pure polyserial submodules
251
Separable modules
253
Submodules of separable modules
256
Direct sums of cyclic modules
259

FINITELY GENERATED AND POLYSERIAL MODULES
173
INVARIANTS AND BASIC SUBMODULES
195
aInvariants
196
aInvariants of equiheight submodules
199
aBasic submodules
201
Modules with trivial ainvariants
205
Notes Problems 18 19
208
RDINJECTIVITY AND PUREINJECTIVITY
209
RDinjective modules
210
Pureinjective modules
214
Pureinjective modules over Prufer domains
220
Pureinjectivity over valuation domains
223
Pureinjective hulls of polyserial modules
228
Notes Problems 2023
231
TORSIONCOMPLETE AND COTORSION MODULES
233
Torsionultracomplete modules
238
Torsion modules of projective dimension one
262
Modules with zero ainvariants
264
Notes Problem 25
268
TORSIONFREE MODULES
269
Completely decomposable modules
273
Finite rank modules over almost maximal valuation domains
277
Rank one dense basic submodules
281
Chains of pure submodules
286
Pure submodules of free modules
292
Slender modules
294
Notes Problem 26
301
References
303
Notation
309
Author Index
312
Subject Index
314
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About the author (1985)

Fuchs is a W.R. IRBY Professor of Mathematics at Tulane University, New Orleans, Louisiana.

Salce is Professor at the University of Padova, Italy.

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